Those who haven't seen the puzzle yet, give it a try at 100 Lockers Puzzle

Student 1 opens all the lockers and Student 2 goes and closes every second locker

and so on... means

Locker 1: No one change the state of 1st locker except student 1 so it is in opened state.

Locker 2: Student 1 opens and Student 2 closes.

Locker 3: Student 1 opens and Student 3 closes.

Locker 4: Student 1 opens, Student 2 closes and Student 4 Opens.

. . .

This is very tedious approach and no one likes to do this physical exercise.But by observing clearly we can conclude

Locker 1 state can be changed byonestudent,

Locker 2 state can be changed bytwostudents,

Locker 3 state can be changed bytwostudents,

Locker 4 state can be changed bythreestudents, ...

which clearly clarifies that.the Lockers are Opened only by the Odd number of students.We can get the count of students who are going to the particular locker by finding the factors of that locker.If we get odd number of factors then that locker is in Opened StateFor Example:

Locker 9 has factors {1,3,9} -> odd count, so it is in opened state.

So there are 10 Lockers which has an odd count. And they are {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.. That is by just calculating theAnother simple way to get how many lockers has an odd number of factorsfrom 1 to 100. So, they are nothing but {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.perfect squares

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